Average Error: 40.6 → 0.7
Time: 48.1s
Precision: 64
Internal Precision: 1408
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{e^{x}}{e^{x} - 1} \le -0.06048001426015734:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{12} \cdot x\right)\\ \mathbf{if}\;\frac{e^{x}}{e^{x} - 1} \le 4644336287.380058:\\ \;\;\;\;\frac{1}{1 - e^{-x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{12} \cdot x\right)\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if (/ (exp x) (- (exp x) 1)) < -0.06048001426015734 or 4644336287.380058 < (/ (exp x) (- (exp x) 1))

    1. Initial program 60.8

      \[\frac{e^{x}}{e^{x} - 1}\]

    2. Taylor expanded around 0 0.1

      \[\leadsto \color{blue}{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{12} \cdot x\right)}\]

    if -0.06048001426015734 < (/ (exp x) (- (exp x) 1)) < 4644336287.380058

    1. Initial program 2.0

      \[\frac{e^{x}}{e^{x} - 1}\]
    2. Using strategy rm

    3. Applied clear-num2.0

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{x} - 1}{e^{x}}}}\]

    4. Applied simplify0.4

      \[\leadsto \frac{1}{\color{blue}{1 - e^{-x}}}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 48.1s)Debug log

herbie shell --seed '#(1743936871 1855164119 3668777427 1254258049 132811564 1366975197)' 
(FPCore (x)
  :name "NMSE section 3.11"
  :pre (!= x 0)
  (/ (exp x) (- (exp x) 1)))